Arguing about pushers vs. pullers is a lot like arguing canards vs. non-canards. Lots of bandwidth and devotees, but not always real substance.
I’m not an engineer and I don’t have a fancy degree in aerodynamics but I did have my flying start in pushers rather than pullers. The first two years I flew as a pilot it was exclusively in pushers.
While pushers (and for that matter, canards) have the distinction of dating back to the Wright Flyers, there was one very basic problem with them that resulted in a lack a popularity in the first third of the 20th Century.
Although it seems incomprehensible to us today, from 1903 through the 1930’s pilots expected things to go wrong on a fairly regular basis. Engine failures - and therefore forced landings - were not a constant feature but they did happen and pilots expected they would experience them from time to time. In a crash, if you were flying a puller the engine was between you and the ground, meaning the airplane (a contraption of wood and cloth mostly in those days) tended to land on the engine (a big hunk of hot metal). In a pusher, YOU were first at the scene and the engine tended to land on the airplane, and you. There were a number of disturbing instances where the forced landing would have been survivable except for the engine having squashed the occupants. In those early years, where you did find pushers the engines tended to be mounted out on the wings so that if they did break free in a crash the pilot wasn’t between them and the ground (in the original Wright Flyer the engine was offset - the pilot lay down beside it, not in front of it.)
Even today, this is taken into consideration. Some designs put up with thrustline difficulties to mount the engine high enough so that in the event of an accident it will tend to sail over the head of the occupants instead of into them. I do know someone where just that happened to them - a sudden stop, with the engine swinging over their head. Which, of course, means the prop is getting closer. He wasn’t hurt, but he did get the crap scared out of him. I can’t repeat what the guy said (except in the Pit) but his next airplane was a puller.
A lot of the other concerns come down to airplane design. There is no such thing as the “perfect airplane”. Some are better suited to some jobs than others, is all.
Some advantages of a pusher:
Improved airflow over the tail means, on average, better handling (when the engine is running…). Yes, there can be problems with handling at low speeds - ditto for t-tails using traditional puller engines. Any aircraft, even a glider, has to be designed to handle the air flows at various speeds. I’m sure if the majority of airplanes were pusher canards we’d be moaning about the “weird airflows” on a puller non-canard.
More accurate ram air for the pitot tube - that means your airspeed indicator tends to be more accurate over a wide range of speeds. Yes, pullers can have accurate airspeed indications, too - but the placement of the pitot can be tricky.
If you’re open cockpit, a pusher (behind you) doesn’t interfere with the air stream, allowing a direct experience of the relative wind and airspeed by the pilot. If you’re flying with minimal instruments (which is how I started) or even NO instruments (they do quit working sometimes) this is a good thing.
Those tail booms - having two booms around the prop makes it less likely some lineman or passenger will accidently walk into a spinning prop. (This is an exceedingly messy accident no one ever wants to see) It’s not impossible to be that foolish, it just cuts down on the likelihood. Which is a good thing.
Some disadvantages:
Pushers tend to pitch down when powered up, and pitch up when powered down or abruptly failing. This is the opposite of the tendencies you want in an aircraft.
Props can and have severed tailbooms. Unless you have a parachute, this is uniformly fatal.
That whole squash-you-like-a-bug problem in crashes. Since aircraft engines of a given horsepower have become smaller and lighter, and attachment devices stronger, this is not as much an issue as it was in 1920 for very small aircraft. However, there is an upper limit beyond which the engine block will tend to go its own way in an accident.
Crap going through the prop. Granted, crap can go through a front-mounted prop, too - but that’s generally problems on the ground with kicked-up debris and/or birds aloft. Pushers have those two problems AND the additional factor that stuff falling off of or out of the airplane can go through the pusher prop. And stuff DOES fall off and out. Had a friend who’s pusher lost an engine bolt, which took a piece of prop blade off as it departed the airplane. This was a Bad Thing (although I’m happy to say he landed safely) Back when I was flying open cockpit pushers making sure stuff was secured was a major major concern.
Those cooling issues:
Back in the bad old days when everything was out in the open this wasn’t as much of an issue. It’s when you start enclosing the engine in a streamlined cowling that it becomes a problem. In honesty, overheating IS a concern with a covered engine and back in the bad old days the difference in keeping the powerplant cool was enough to reinforce the move from pushers to pullers already started by the crash issues. With the modern ultralight and homebuilt pushers I have seen water-cooling and radiators used to solve that problem. Yes, it adds weight but remember that the modern small aircraft engine is much more efficient than what was used 80 years ago. The 1903 Wright engine was what? 12 horsepower? Weighed a couple hundred pounds? A 50 hp engine is available that is small enough for one person to lift (alright, it’s kinda heavy, but…). Ultralights and homebuilts can lift a payload equal to their own empty weight off the ground and still fly better that what was aloft in 1920. These days, we can afford the extra couple pounds to keep the engine cool, which they couldn’t do 80 years ago.
Those airflow issues:
Here we’re getting into areas where I don’t have a lot of expertise. I can see where airflow can be an issue, that makes sense to my pilot’s eye. But airflow issues are always present - the airflow over the fuselage always affects the tail surfaces, which are vital in most aircraft (canards being the exception) and must therefore be accounted for regardless of engine location.
My experience has been that, with the exception of the pitch tendencies already mentioned, there’s not a heck of a lot of difference from the pilot’s viewpoint. All this squirrelliness I hear rumors about is just that - rumors. Yes, there are some horrifically poor designs in the ultralight/microlight/homebuilt category - but that applies to pullers as well as pushers. The ones I flew were pretty docile in the air, arguably more so than the Cessna 150, which is a puller and not generally regarded as a high performance tricky airplane. Maybe it’s a matter of having the kinks worked out of the design by now.
A word about the Cessna Skymaster, the C-337 - I’m not an authority but we do have one at my local field. The “critical engine” is, in fact, the rear engine. On the rear engine alone it’s quite controllable and can even produce a useful (although not wonderful) rate of climb. On just the front engine… well, the pilot owner says don’t expect to maintain level flight and it’s just not fun. Now, why that is so I don’t know, but at least in THAT airplane the pusher engine is more important/useful than the puller.
Oh, and parachutes - I know folks who have jumped out of plenty of pushers. This doesn’t seem to be an issue. Any time you jump from an airplane you have to jump away from it as well - hitting the tail of a puller plane can be just as deadly as hitting the prop of a pusher.